Coupling of quantum well states and phonons in thin multilayer Pb films on Si(111)

Abstract

Density functional theory calculations for the electronic and phononic band structures of Pb/Si(111) thin films with a thickness of 4 and 5 monolayers (ML) are performed. We employ a Si(111)$(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})$ unit cell to model the Pb films including the Si substrate, and identify quantum well (QW) states for film thicknesses between 3 and 6 ML. The calculations show that the quantum-confined state closest to the Si band gap acquires the character of a quantum well resonance with a major part of its wave function extending into the Si(111) substrate. This finding explains the unusually low dispersion of this state and its lacking sensitivity to phonons in the 5 ML Pb film. Moreover, several unoccupied QW states are identified in the calculations and are assigned to previously observed features in structurally simpler freestanding Pb films. The calculated phonon band structures of the Pb/Si(111)$(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})$ films display stiff surface phonon modes in the 2.3--2.5 THz range. The electron-phonon coupling strength in the quantum-confined states is addressed by means of deformation-potential theory using the calculated atomic displacements of $\mathrm{\ensuremath{\Gamma}}$-point phonons. It is found that both the acoustic shear deformation potential as well as the optical deformation potentials of unoccupied QW states are sizable. Comparing the results for 4 and 5 ML Pb films, we conclude that the optical deformation potentials are generally larger for the 4 ML film. The occupied QW resonance in the 5 ML Pb film shows weak electron-phonon coupling, in qualitative agreement with the small experimentally observed lifetime broadening of this state. Our results form the basis for addressing the role of electron-phonon scattering for the lifetime of unoccupied QWs acting as intermediate states in two-photon photoemission from Pb/Si(111) films.